On the existence of solutions for systems of generalized vector quasi-variational equilibrium problems in abstract convex spaces with applications

Chengqing Pan; Haishu Lu; Chengqing Pan; Haishu Lu

doi:10.3934/math.20241447

AIMS Mathematics

2024, Volume 9, Issue 11: 29942-29973. doi: 10.3934/math.20241447

Previous Article Next Article

Research article

On the existence of solutions for systems of generalized vector quasi-variational equilibrium problems in abstract convex spaces with applications

Chengqing Pan ^1,2,
Haishu Lu ^{3
,
,}

1.
School of Economics and Management, China University of Mining and Technology, Xuzhou 221116, China
2.
Office of Discipline Inspection Commission, Jiangsu University of Technology, Changzhou 213001, China
3.
School of Economics, Jiangsu University of Technology, Changzhou 213001, China

Received: 29 July 2024 Revised: 28 September 2024 Accepted: 11 October 2024 Published: 22 October 2024
MSC : 47H04, 47H10, 91A10

In this paper, we first introduced systems of generalized vector quasi-variational equilibrium problems as well as systems of vector quasi-variational equilibrium problems as their special cases in abstract convex spaces. Next, we established some existence theorems of solutions for systems of generalized vector quasi-variational equilibrium problems and systems of vector quasi-variational equilibrium problems in non-compact abstract convex spaces. Furthermore, we applied the obtained existence theorem of solutions for systems of vector quasi-variational equilibrium problems to solve the existence problem of Nash equilibria for noncooperative games. Then, as applications of the existence result of Nash equilibria for noncooperative games, we studied the existence of weighted Nash equilibria and Pareto Nash equilibria for multi-objective games. The results derived in this paper extended and unified the primary findings presented by some authors in the literature.
Citation: Chengqing Pan, Haishu Lu. On the existence of solutions for systems of generalized vector quasi-variational equilibrium problems in abstract convex spaces with applications[J]. AIMS Mathematics, 2024, 9(11): 29942-29973. doi: 10.3934/math.20241447

Related Papers:

Abstract

In this paper, we first introduced systems of generalized vector quasi-variational equilibrium problems as well as systems of vector quasi-variational equilibrium problems as their special cases in abstract convex spaces. Next, we established some existence theorems of solutions for systems of generalized vector quasi-variational equilibrium problems and systems of vector quasi-variational equilibrium problems in non-compact abstract convex spaces. Furthermore, we applied the obtained existence theorem of solutions for systems of vector quasi-variational equilibrium problems to solve the existence problem of Nash equilibria for noncooperative games. Then, as applications of the existence result of Nash equilibria for noncooperative games, we studied the existence of weighted Nash equilibria and Pareto Nash equilibria for multi-objective games. The results derived in this paper extended and unified the primary findings presented by some authors in the literature.

References

[1]	Q. H. Ansari, W. K. Chan, X. Q. Yang, The system of vector quasi-equilibrium problems with applications, J. Global Optim., 29 (2004), 45–57. https://doi.org/10.1023/B:JOGO.0000035018.46514.ca doi: 10.1023/B:JOGO.0000035018.46514.ca
[2]	Q. H. Ansari, Z. Khan, System of generalized vector quasi-equilibrium problems with applications, Narosa Publishing House, 2004.
[3]	L. J. Lin, Systems of generalized quasivariational inclusions problems with applications to variational analysis and optimization problems, J. Global Optim., 38 (2007), 21–39. https://doi.org/10.1007/s10898-006-9081-5 doi: 10.1007/s10898-006-9081-5
[4]	L. J. Lin, L. F. Chen, Q. H. Ansari, Generalized abstract economy and systems of generalized vector quasi-equilibrium problems, J. Comput. Appl. Math., 208 (2007), 341–353. https://doi.org/10.1016/j.cam.2006.10.002 doi: 10.1016/j.cam.2006.10.002
[5]	L. J. Lin, Q. H. Ansari, Systems of quasi-variational relations with applications, Nonlinear Anal., 72 (2010), 1210–1220. https://doi.org/10.1016/j.na.2009.08.005 doi: 10.1016/j.na.2009.08.005
[6]	S. Al-Homidan, Q. H. Ansari, S. Schaible, Existence of solutions of systems of generalized implicit vector variational inequalities, J. Optim. Theory Appl., 134 (2007), 515–531. https://doi.org/10.1007/s10957-007-9236-7 doi: 10.1007/s10957-007-9236-7
[7]	M. Patriche, New results on systems of generalized vector quasi-equlibrium problems, Taiwanese J. Math., 19 (2015), 253–277. https://doi.org/10.11650/tjm.19.2015.4098 doi: 10.11650/tjm.19.2015.4098
[8]	J. W. Peng, H. W. J. Lee, X. M. Yang, On system of generalized vector quasiequilibrium problems with set-valued maps, J. Global Optim., 36 (2006), 139–158. https://doi.org/10.1007/s10898-006-9004-5 doi: 10.1007/s10898-006-9004-5
[9]	J. W. Peng, S. Y. Wu, The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems, Optim. Lett., 4 (2010), 501–512. https://doi.org/10.1007/s11590-010-0179-9 doi: 10.1007/s11590-010-0179-9
[10]	Z. Lin, The study of the system of generalized vector quasi-equilibrium problems, J. Global Optim., 36 (2006), 627–635. https://doi.org/10.1007/s10898-006-9031-2 doi: 10.1007/s10898-006-9031-2
[11]	S. H. Hou, H. Yu, G. Y. Chen, On system of generalized vector variational inequalities, J. Global Optim., 40 (2008), 739–749. https://doi.org/10.1007/s10898-006-9112-2 doi: 10.1007/s10898-006-9112-2
[12]	A. Farajzadeh, B. S. Lee, S. Plubteing, On generalized quasi-vector equilibrium problems via scalarization method, J. Optim. Theory Appl., 168 (2016), 584–599. https://doi.org/10.1007/s10957-015-0772-2 doi: 10.1007/s10957-015-0772-2
[13]	N. V. Hung, LP well-posed controlled systems for bounded quasi-equilibrium problems and their application to traffic networks, J. Comput. Appl. Math., 401 (2022), 113792. https://doi.org/10.1016/j.cam.2021.113792 doi: 10.1016/j.cam.2021.113792
[14]	S. Al-Homidan, Q. H. Ansari, Fixed point theorems on product topological semilattice spaces, generalized abstract economies and systems of generalized vector quasi-equilibrium problems, Taiwanese J. Math., 15 (2011), 307–330. https://doi.org/10.11650/twjm/1500406176 doi: 10.11650/twjm/1500406176
[15]	S. Plubtieng, T. Thammathiwat, Existence of solutions of systems of generalized implicit vector quasi-equilibrium problems in $G$-convex spaces, Comput. Math. Appl., 62 (2011), 124–130. https://doi.org/10.1016/j.camwa.2011.04.059 doi: 10.1016/j.camwa.2011.04.059
[16]	X. P. Ding, The generalized game and the system of generalized vector quasi-equilibrium problems in locally $FC$-uniform spaces, Nonlinear Anal., 68 (2008), 1028–1036. https://doi.org/10.1016/j.na.2006.12.003 doi: 10.1016/j.na.2006.12.003
[17]	X. P. Ding, Collective fixed points, generalized games and systems of generalized quasi-variational inclusion problems in topological spaces, Nonlinear Anal., 73 (2010), 1834–1841. https://doi.org/10.1016/j.na.2010.05.018 doi: 10.1016/j.na.2010.05.018
[18]	S. Park, Elements of the KKM theory on abstract convex spaces, J. Korean Math. Soc., 45 (2008), 1–27. https://doi.org/10.4134/JKMS.2008.45.1.001 doi: 10.4134/JKMS.2008.45.1.001
[19]	C. D. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl., 156 (1991), 341–357. https://doi.org/10.1016/0022-247X(91)90402-L doi: 10.1016/0022-247X(91)90402-L
[20]	C. D. Horvath, J. V. L. Ciscari, Maximal elements and fixed points for binary relations on topological ordered spaces, J. Math. Econom., 25 (1996), 291–306. https://doi.org/10.1016/0304-4068(95)00732-6 doi: 10.1016/0304-4068(95)00732-6
[21]	S. Park, On the von Neumann-Sion minimax theorem in KKM spaces, Appl. Math. Lett., 23 (2010), 1269–1273. https://doi.org/10.1016/j.aml.2010.06.011 doi: 10.1016/j.aml.2010.06.011
[22]	S. Park, The KKM principle in abstract convex spaces: equivalent formulations and applications, Nonlinear Anal., 73 (2010), 1028–1042. https://doi.org/10.1016/j.na.2010.04.029 doi: 10.1016/j.na.2010.04.029
[23]	S. Park, New generalizations of basic theorems in the KKM theory, Nonlinear Anal., 74 (2011), 3000–3010. https://doi.org/10.1016/j.na.2011.01.020 doi: 10.1016/j.na.2011.01.020
[24]	Q. H. Ansari, S. Schaible, J. C. Yao, The system of generalized vector equilibrium problems with applications, J. Global Optim., 22 (2002), 3–16. https://doi.org/10.1023/A:1013857924393 doi: 10.1023/A:1013857924393
[25]	G. Kassay, M. Miholca, N. T. Vinh, Vector quasi-equilibrium problems for the sum of two multivalued mappings, J. Optim. Theory Appl., 169 (2016), 424–442. https://doi.org/10.1007/s10957-016-0919-9 doi: 10.1007/s10957-016-0919-9
[26]	A. Capǎtǎ, On vector quasi-equilibrium problems via a Browder-type fixed-point theorem, Bull. Malays. Math. Sci. Soc., 46 (2023), 14. https://doi.org/10.1007/s40840-022-01397-8 doi: 10.1007/s40840-022-01397-8
[27]	X. P. Ding, Existence of Pareto equilibria for constrained multiobjective games in $H$-space, Comput. Math. Appl., 39 (2000), 125–134. https://doi.org/10.1016/S0898-1221(00)00092-4 doi: 10.1016/S0898-1221(00)00092-4
[28]	H. S. Lu, Q. W. Hu, A collectively fixed point theorem in abstract convex spaces and its applications, J. Funct. Space, 2013 (2013), 517469. http://doi.org/10.1155/2013/517469 doi: 10.1155/2013/517469
[29]	M. Patriche, Existence of equilibrium for multiobjective games in abstract convex spaces, Math. Rep., 2 (2014), 243–252.
[30]	S. Y. Wang, Existence of a Pareto equilibrium, J. Optim. Theory Appl., 79 (1993), 373–384. https://doi.org/10.1007/BF00940586 doi: 10.1007/BF00940586
[31]	H. S. Lu, X. Q. Liu, R. Li, Upper semicontinuous selections for fuzzy mappings in noncompact $WPH$-spaces with applications, AIMS Math., 7 (2022), 13994–14028. http://doi.org/10.3934/math.2022773 doi: 10.3934/math.2022773

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)